2 for dyads and 3 for triads However not every tensor can be synthesized as a

2 for dyads and 3 for triads however not every tensor

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2 for dyads and 3 for triads. However, not every tensor can be synthesized as a product of [45] As indicated before, there are cases of tensors which are not uniformly dimensioned, and in some cases these tensors can be regarded as the result of an outer product of lower rank tensors. [46] Regarding the associativity of direct multiplication, there seems to be cases in which this operation is not associative. The interested reader is advised to refer to the research literature on this subject.
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3.4 Contraction 87 lower rank tensors. Multiplication of a tensor by a scalar (refer to § 3.2) may be regarded as a special case of direct multiplication. 3.4 Contraction The contraction operation of a tensor of rank > 1 is to make two free indices identical , by unifying their symbols, and perform summation over these repeated indices, e.g. A j i contraction --------→ A i i (108) A jk il contraction on jl ------------→ A mk im (109) Contraction results in a reduction of the rank by 2 since it implies the annihilation of two free indices. Therefore, the contraction of a rank-2 tensor is a scalar, the contraction of a rank-3 tensor is a vector, the contraction of a rank-4 tensor is a rank-2 tensor, and so on. For general non-Cartesian coordinate systems, the pair of contracted indices should be different in their variance type, i.e. one upper and one lower. Hence, contraction of a mixed tensor of type ( m, n ) will, in general, produce a tensor of type ( m - 1 , n - 1 ). A tensor of type ( p, q ) can, therefore, have p × q possible contractions, i.e. one contraction for each combination of lower and upper indices. A common example of contraction is the dot product operation on vectors (see § Dot Product) which can be regarded as a direct multiplication (refer to § 3.3) of the two vectors, which results in a rank-2 tensor, followed by a contraction. Also, in matrix algebra, taking the trace of a square matrix, by summing its diagonal elements, can be considered as a contraction operation on the rank-2 tensor represented by the matrix, and hence it yields the trace which is a scalar. Conducting a contraction operation on a tensor results into a tensor . Similarly, the
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3.5 Inner Product 88 application of a contraction operation on a relative tensor (see § 2.6.3) produces a relative tensor of the same weight as the original tensor. 3.5 Inner Product On taking the outer product (refer to § 3.3) of two tensors of rank 1 followed by a contraction (refer to § 3.4) on two indices of the product, an inner product of the two tensors is formed. Hence, if one of the original tensors is of rank- m and the other is of rank- n , the inner product will be of rank- ( m + n - 2 ). In the symbolic notation of tensor calculus, the inner product operation is usually symbolized by a single dot between the two tensors, e.g. A · B , to indicate the contraction operation which follows the outer multiplication.
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  • Summer '20
  • Rajendra Paramanik
  • Tensor, Coordinate system, Polar coordinate system, Coordinate systems

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